Tracking Control for Triple-Integrator and Quintuple-Integrator Systems with Single Input Using Zhang Neural Network with Time Delay Caused by Backward Finite-Divided Difference Formulas for Multiple-Order Derivatives

Abstract

Tracking control for multiple-integrator systems is regarded as a fundamental problem associated with nonlinear dynamic systems in the physical and mathematical sciences, with many applications in engineering fields. In this paper, we adopt the Zhang neural network method to solve this nonlinear dynamic problem. In addition, in order to adapt to the requirements of real-world hardware implementations with higher-order precision for this problem, the multiple-order derivatives in the Zhang neural network method are estimated using backward finite-divided difference formulas with quadratic-order precision, thus producing time delays. As such, we name the proposed method the Zhang neural network method with time delay. Moreover, we present five theorems to describe the convergence property of the Zhang neural network method without time delay and the quadratic-order error pattern of the Zhang neural network method with time delay derived from the backward finite-divided difference formulas with quadratic-order precision, which specifically demonstrate the effect of the time delay. Finally, tracking controllers with quadratic-order precision for multiple-integrator systems are constructed using the Zhang neural network method with time delay, and two numerical experiments are presented to substantiate the theoretical results for the Zhang neural network methods with and without time delay.

Well-Posedness, Dynamics, and Control of Nonlinear Differential System with Initial-Boundary Value

“…It is proved that there is a unique bounded global solution to the impulsive delay differential systems with stochastic control, and the boundedness of this global solution is illuminated by numerical simulations. In reference [14], a second-order precision tracking controller for multi-integrator systems was constructed using the delayed Zhang neural network method, and numerical experiments were conducted to verify the theoretical results of the Zhang neural network method.…”

“…The series of papers in this Special Issue reflect the application of dynamic system theory in finance, neural networks, and other control systems [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. For example, the authors of [1] studied the stability of PSS in hyperchaotic financial systems.…”

“…On the other hand, the stability and synchronization control of dynamic systems are also among the main contents of cybernetics, and they are part of well-posed problems. Fifteen papers [1][2][3][4][5][6][7][8][9][10][11][12][13][14] in this Special Issue deal with the stabilization and synchronization control of initial boundary value problems of partial differential equations or ordinary differential equations, and jointly complete the original work plan of this Special Issue. The papers in this Special Issue can be divided according to the following scheme considering their main purposes:…”

High-Order Modeling, Zeroing Dynamics Control, and Perturbations Rejection for Non-Linear Double-Holding Water Tank

“…In recent years, zeroing dynamics (ZD) has been proposed and investigated for the solution of time-varying problems. Due to its advantages in terms of convergence and accuracy, the ZD method has been successfully extended to various areas, including automatic control, robotics, and numerical computation [8,[28][29][30][31][32][33][34][35]. As a powerful method, ZD has been substantiated for the tracking of non-linear systems.…”